Use the following sample dialog to inspire a similar exchange between you and your students where you play the part of Mike, suggesting ideas of actions you could perform on both sides of an equation that would not predictably preserve the solution set of the original equation. Start by asking students to summarize what they have been studying over the last two lessons and then make Mike’s first suggestion. Be sure to provide more than one idea for things that could be done to both sides of an equation that might result in solutions that are not part of the solution set for the original equation, and conclude with an affirmation that you can try anything, but you will have to check to see if your solutions work with the original equation.

Fergus says, “Basically, what I’ve heard over the last two lessons is that whatever you do to the left side of the equation, do the same thing to the right side. Then solutions will be good.”

Lulu says, “Well, we’ve only said that for the properties of equality – adding quantities and multiplying by non-zero quantities. (And associative, commutative, and distributive properties too.) Who knows if it is true in general?”

Mike says, “Okay … Here’s an equation:

x12=13

If I follow the idea, “Whatever you do to the left, do to the right as well,” then I am in trouble. What if I decide to remove the denominator on the left and also remove the denominator on the right. I get
x=1
. Is that a solution?”

Fergus replies, “Well, that is silly. We all know that is a wrong thing to do. You should multiply both sides of that equation by
12
. That gives
x=4
, and that does give the correct solution.”

Lulu says, “Okay Fergus, you have just acknowledged that there are some things we can’t do! Even if you don’t like Mike’s example, he’s got a point.”

Mike or another student says, “What if I take your equation and choose to square each side. This gives

x2144=19

Multiplying through by
144
gives
x2=1449=16
, which has solutions
x=4
AND
x=-4
.”

Fergus responds, “Hmmm. Okay I do see the solution
x=4
, but the appearance of
x=-4
as well is weird.”

Mike says, “Lulu is right. Over the past two days we have learned that using the commutative, associative, and distributive properties, along with the properties of equality (adding and multiplying equations throughout) definitely DOES NOT change solution sets. BUT if we do anything different from this we might be in trouble.”

Lulu continues, “Yeah! Basically when we start doing unusual operations on an equation, we are really saying that **IF** we have a solution to an equation, then it should be a solution to the next equation as well. BUT remember, it could be that there was no solution to the first equation anyway!”

Mike says, “So feel free to start doing weird things to both sides of an equation if you want (though you might want to do sensible weird things!), but all you will be getting are possible CANDIDATES for solutions. You are going to have to check at the end if they really are solutions.”